Question Detail

This integral can be solved using partial fraction decomposition.1. Factor the denominator: x² + 6x - 7 = (x - 1)(x + 7).2. Set up the partial fraction decomposition: 5x / [(x - 1)(x + 7)] = A/(x - 1) + B/(x + 7)3. Multiply by the common denominator to clear fractions: 5x = A(x + 7) + B(x - 1)4. Solve for A and B: * Set x = 1: 5(1) = A(1 + 7) + B(0) => 5 = 8A => A = 5/8 * Set x = -7: 5(-7) = A(0) + B(-7 - 1) => -35 = -8B => B = 35/85. Substitute A and B back into the integral: ∫ [5/(8(x - 1)) + 35/(8(x + 7))] dx6. Integrate term by term: = (5/8) ∫ (1/(x - 1)) dx + (35/8) ∫ (1/(x + 7)) dx = (5/8)ln|x - 1| + (35/8)ln|x + 7| + COptions A and B are identical and match this result. Since B is marked, we choose B.